Computed imaging systems are now widely employed in a variety of applications, including medicine, astronomy and terrain analysis. Imaging modalities used for such applications include computer tomography, magnetic resonance imaging, ultrasound, synthetic aperture radar and radio astronomy.
Of particular interest here, synthetic aperture radars (SARs) have been employed to produce high-quality images of the earth's terrain. Such images are obtained by overhead transmission/receipt of pulses of microwave energy at a predetermined frequency. In this regard SARs provide particular functionality due to their capability to image in darkness and to penetrate cloud-cover. Further, because SAR systems use a form of coherent illumination, SAR systems are capable of transducing the complex reflectivity of terrain within an imaged region. In such applications, the reflectivity function is modulated by phase terms that are dependent upon the imaging system geometry. As a result, when two SAR images are made of an imaged region, it is possible to interfere the two complex images in such a way as to cancel the scene reflectivity which is common to both images and recover the information that contains the scene topography transduced by the image-domain phase data. Such systems may be generally referred to as interferometric synthetic aperture radar (IFSAR) systems.
IFSAR systems, both aircraft and space borne, have been used with moderate success to date to provide terrain height for regions on the earth's surface. Such systems may consist of a single vehicle having one radar transmitter and two spaced receive antennas mounted thereupon, wherein two complex images of an image region may be obtained upon a single pass of the vehicle over an imaged region. Alternatively, the system may comprise one or more vehicles that each have one radar transmitter and one receive antenna mounted thereupon, wherein two complex images are obtained by passing over an imaged region twice. In either case, two complex images of the same region are formed. After acquisition, the images may be registered such that the phase differences between corresponding image pixels, or data samples, may be extracted to form an interferogram. As will be appreciated, the phase differences reflected by the interferogram are wrapped. That is, the phase differences are ambiguous module two pi (2,.pi.).
In order to derive height information from the interferogram, the wrapped phase differences must be unwrapped and corresponding integration constants must be determined. As such, phase unwrapping should be completed in a manner that resolves the 2.pi. ambiguities so that unambiguous terrain heights can be assigned to the phase values. In addressing such task, it has been recognized that the imaged terrain cannot be of a nature that yields phase values that exceed the Nyquist rate without adversely impacting the accuracy of results.
Specifically, adjacent sample-to-sample phase differences of unwrapped interferometric data should be no more than 180.degree.. Such limitation can become problematic when the imaged region comprises steep pastoral terrain (e.g., near vertical natural geographical features) or cultural features (e.g., man-made structures such as buildings). When such features are present, phase unwrapping may result in inconsistent data that renders the entire height estimation unreliable.
To understand such inconsistencies, consider a closed path consisting of one step forward, a step to the left, a step to the left again, and then a final step to the left. After the four steps, one should active back at the starting point. Interferometric differential phase data is supposed to represent terrain height, but in situations that present the above-mentioned problem (i.e., adjacent samples whose phase difference is &gt;180 .degree.), it is possible that the sum of phase differences around a closed, four-point path in the interferogram is non-zero. Such a result would indicate that if one converted the phase differences to height differences and summed them around the path, one would not arrive back at the starting height.
Existing phase unwrapping algorithms are of two general types: least squares and path following. Least-squares algorithms determine the phase surface which best fits the ensemble of pixel-to-pixel phase differences over the entire interferogram. If inconsistencies of the above-noted nature are present, the least-squares process attempts to minimize their deleterious effects by minimizing the residual fitting error. Path-following algorithms, on the other hand, numerically integrate the pixel-to-pixel phase differences over the interferogram, in the process either avoiding or minimizing inconsistencies by selecting closed paths where error is minimized.
Systems based on these methods have not been able to meet the requirements of many potential applications. To date, IFSAR systems have achieved accuracies of a few meters on the average. But because of "errors" in the unwrapping algorithms (i.e., due to the noted inconsistencies), accuracies may be excellent in one region of the image and quite poor in another, and one has no way of knowing which regions are good and which are bad. Further, future systems are desired which can provide sub-meter accuracies with a high degree of assurance. The noted phase unwrapping techniques are not up to the task.
In this regard, both least-squares and path following algorithms can be undermined by terrain characteristics that are relatively common. By way of example, if the number of inconsistencies in an imaged region is large, the least-squares solution is impractically crude because the effects of the bad data are smeared throughout the image. If one has a priori knowledge of regions of defective data, then a weighted least-squares can be used which assigns low, or zero weight to poor data. To date, however, no one has developed a robust way to determine appropriate weights from the interferogram.
Path-following methods are also undermined by large numbers of inconsistencies because automated techniques for finding satisfactory integration paths fail, and the underlying algorithm cannot complete the unwrapping process. One does not know in advance when such failures will occur.
A final difficulty with existing phase unwrapping methods relates to the need to incorporate large numbers of tie points (i.e., locations within the imaged region having known heights) into the algorithm employed in the corresponding systems. To ensure the best possible accuracy, one should incorporate as many known height tie points as possible.
Neither least-squares nor path-following systems lend themselves to incorporating dense grids of tie points, and this will be needed in future systems which aim for sub-meter accuracies.